Mathematical Sciences: Analysis and Numerical Methods in Stochastic Optimization

9529738这个随机优化项目包括四个部分：奇异摄动马氏链的性质的研究；全局随机优化算法的发展；被动随机逼近过程的研究；制造系统中随机优化的数值方法的设计。第一部分主要研究奇摄动非平稳马氏链和弱相互作用和强相互作用的摄动链的渐近性质，以及相关控制问题的渐近最优性。第二部分包括全局随机优化算法的渐近性质的发展。利用弱收敛方法，建立了收敛速度和收敛速度，并证明了适当尺度的序列收敛于适当的随机微分方程解。第三部分，将随机逼近方法与核估计相结合，得到被动随机逼近算法。这里的目标是建立算法的收敛并获得误差估计。第四部分包括求解一类分段确定性马尔可夫链的鲁棒控制问题的数值步骤，以及在多机器制造系统的WIP(Work In Process)策略下逼近最优阈值的随机梯度下降算法。得到了算法的收敛和渐近性质。整个研究项目将包括两个部分：分析和模拟。主要目标是加强对基本系统的渐近性质的基本了解，并开发合理和可行的算法。这项提案由四个密切相关的主题组成。这是将随机优化理论与应用相结合的一种尝试。重点是通过分析和数值实验来开发有效的数值方法。第一个主题涉及易受快速和随机变化影响的系统。所得结果对分层决策、生产计划、通信排队网络和系统可靠性等方面的应用具有一定的参考价值。为了满足对用于全局优化的高效数值程序的日益增长的需求，该项目的第二部分侧重于在必须考虑数据中的随机误差的情况下开发数值算法。所得结果适用于估计、辨识和优化问题中不断扩大的应用范围。本计划第三部分的研究源于化学工程中的稳态估计和检测问题(对于连续搅拌的槽和对于二元精馏塔)。重点是数值算法的设计和实现，预计这些算法在目标识别、跟踪、系统故障检测、信号处理等相关领域也将有各种应用。该计划的第四部分涉及不可靠制造系统的优化方法。通过强调系统的稳定性而不是单纯的最优性，开发了一类生产计划的鲁棒控制问题的数值方法。在日本汽车业最初推动的当今流行的看板政策下，将开发一种数值程序，该程序将为制造系统提供一种系统地寻找最佳看板数量的方法。***

9529738 Yin This project in stochastic optimization consists of four parts: an examination of the properties of singularly perturbed Markov chains; a development of global stochastic optimization algorithms; a study of passive stochastic approximation procedures; and the design of numerical methods for stochastic optimization in manufacturing systems. The first part is intended to derive asymptotic properties of singularly perturbed nonstationary Markov chains and perturbed chains with weak and strong interactions, and to obtain asymptotic optimality in related control problems. The second part comprises the development of asymptotic properties of global stochastic optimization algorithms. By using weak convergence methods, convergence and rates of convergence are to be established, and the convergence of suitably scaled sequences to the solutions of appropriate stochastic differential equations is to be ascertained. In the third part, by combining stochastic approximation methods with kernel estimators, passive stochastic approximation algorithms are to be obtained. Here the goal is to establish convergence of the algorithms and to obtain error estimates. The fourth part encompasses numerical procedures for solving a class of robust control problems of piecewise deterministic Markov chains, and the development of stochastic gradient descent algorithms for approximating the optimal threshold values under the WIP (work in progress) policies for multi-machine manufacturing systems. Convergence and asymptotic properties of the algorithms are to be obtained. The entire research project will include two components: analysis and simulation. The main objectives are to enhance basic understanding of the asymptotic properties of the underlying systems and to develop sound and feasible algorithms. This proposal consists of four closely related topics. It is a bridge-building attempt to link theory and application in stochastic optimization. Emphasis is on the development of efficient numerical methods via analysis and numerical experimentation. The first topic deals with systems that are subject to rapid and random variations. The results obtained are intended be useful for applications involving hierarchical decision making, production planning, queuing networks in communication, and system reliability. To meet the increasing demand for efficient numerical procedures for global optimization, the second part of the project focuses on the development of numerical algorithms when random errors in the data have to be taken into account. The results are applicable to an ever expanding range of applications in estimation, identification and optimization problems. The research undertaken in the third part of this program derives from steady state estimation and detection problems in chemical engineering (for a continuously stirred tank and for a binary distillation column). The emphasis is on the design and implementation of numerical algorithms, which are expected to also have a variety of applications in target recognition, tracking, system failure detection, signal processing and related fields. The fourth part of the program is concerned with optimization methods for unreliable manufacturing systems. By emphasizing system stability rather than optimality alone, numerical methods for a class of robust control problems of production planning are to be developed. Under nowadays popular Kanban policies, which were originally promoted by the Japanese auto industry, a numerical procedure that will provide a systematic way of finding the optimal number of Kanbans for manufacturing systems is to be developed.. ***